Static and dynamic size optimization of nonlinear continuous structure based on fully stressed design

Abstract

In order to realize the fully stressed criterion, that is, the stress of the nonlinear engineering structure is uniform along the height direction under the action of wind load or ground motion, the section size of the continuous structure is optimized based on mechanical analysis. The engineering structure is simplified as a continuous cantilever with variable cross-section, and its material constitution is nonlinear. The wind load and ground motion are represented by three kinds of horizontal static loads: uniform load, inverted triangle load, and inertia force related load. In order to meet the needs of different projects, hollow circular section and box section are designed, respectively. By establishing different expressions of section moment, the optimal section size distribution is solved. Then the optimal stiffness distribution of nonlinear structure is proposed. The correctness of the theory is verified by the finite element method. The results are suitable for the elastic and elastic-plastic stages of the structure, and are effective for both static loads and dynamic actions. The optimal section size distribution and structural shape are different under different loads. Finally, a practical design example is shown.

Keywords

fully stressed criterion nonlinear continuous structures optimal stiffness distribution section size optimization seismic action

Introduction

Structural optimization design refers to the design method of obtaining the optimal scheme according to certain optimization objectives (such as the lightest weight, the lowest cost, the maximum stiffness, etc.) (Ahrari and Deb, 2016; Gil-Martín et al., 2017; Guan et al., 1999; Hernández-Montes et al., 2018; San et al., 2020). The methods of structural optimization include criterion method, mathematical programing method, and intelligent algorithm method (Bruyneel et al., 2012; Dilgen et al., 2019; Gholizadeh and Mohammadi, 2017; Xu et al., 2005). Criterion method is the most basic method to solve structural optimization problems, which puts forward mechanical criteria (such as fully stressed criterion, stiffness criterion, energy criterion, etc.) according to the engineering requirements (Ahrari and Deb, 2016; Potra and Simiu, 2009).

In recent years, the research on continuum topology optimization has been deep and abundant, and some applications have been achieved (Chan and Zou, 2004; Chan et al., 1994, 2010; Zou and Chan, 2005). However, for practical engineering structures, the region boundary needs to maintain smooth continuity conditions, and the optimized shape boundary needs to be producible enough. Topology optimization pays too much attention to material distribution, but does not fully consider the limitation of actual conditions. Its load conditions are monotonous, so its engineering feasibility is weak. Therefore, it is of great engineering value to explore the theoretical optimization of continuum section and shape.

In addition, fully stressed design criterion is the most concerned among various optimization criteria. Its main idea is to make every component of the structure reach the ultimate bearing capacity under the specified loading mode, so that the strength of the component can be fully utilized (Hajirasouliha et al., 2004; Mergos, 2018). In the traditional study of structural cross-section size optimization, fully stressed design criteria should be given priority to avoid local deformation caused by stress concentration or weak story. In the present research, this method are usually used to optimize structure assuming that the structure is always elastic. Under these conditions, the fully stressed design criterion is equivalent to the equal strength criterion or the equal stress criterion. A lot of achievements have been made in equal stress optimization of components or simple structures. For example, the fully stressed optimization of simply supported beams under simple loads has been mature (Ahlawat and Ramaswamy, 2004; Truman and Cheng, 1990).

Due to engineering structures are usually subjected to earthquake, wind and other stochastic loads, especially when the excitation amplitude is large, the material of the structure is likely to enter elastic-plastic state. If the stress of the structure is uniform when it is close to the critical failure, the damage of each part will be similar. Therefore, the fully stressed optimization results can also provide the initial optimization scheme for the uniform damage design. However, most current studies have not fully considered elastic-plastic stage of structural materials, that is, fully stressed optimization design has not really been realized. Therefore, it is necessary to optimize nonlinear structures based on fully stressed criterion. The optimization effect is shown in Figure 1.

Figure 1.

Optimization effect.

In summary, there are still several shortcomings in engineering structure optimization. At first, the equal stress concept are usually used in components optimization under simple loads. The overall structure optimization under complex loads is not intensive and comprehensive enough. Secondly, the mechanics mechanism research on optimal section size distribution of continuous structure is inadequate, which cannot provide strict and accurate analytical formulas for structural optimization design. Finally, the research on fully stressed design of nonlinear material structures is rare, and it cannot meet the requirement of actual engineering structures.

In this paper, engineering structures such as continuous tower structure and high-rise building are simplified into nonlinear continuous variable cross-section cantilever. By establishing different expressions of bending moments, the analytical solution of optimal section size is obtained. The quadratic polynomial obtained by fitting the experimental data is used to characterize the nonlinear constitutive law of the material. The optimal section size distribution formulas of circular, rectangular, hollow circular and box section cantilevers under uniform load, inverted triangle load and inertia force related load are derived respectively. The complete analytical solution and the optimal stiffness distribution analytical solution are obtained. Finally, the above theoretical methods are fully verified by finite element method.

It must be pointed out that due to the variety of actual high-rise building structural members and the complexity of structural forms, it is difficult to obtain the analytical solution of optimal lateral stiffness distribution from the mechanical optimization. Iterative algorithm is needed for depth optimization design. Therefore, based on the mechanical theory, this paper analyses the simplified model and obtains the analytical solutions of the optimal section size distribution. The results can be used to optimize continuous cantilever structure such as tower structure. At the same time, it can provide the initial optimization solution for the optimization design of engineering structures and improve the efficiency.

Optimal section size distribution of nonlinear structures under differentload forms

Structure section form, load model, and material constitutive

The displacement and damage of engineering structures under external excitation are usually closely related to its stiffness and mass distribution. Structural stiffness after reasonable design usually varies continuously along the height direction. Continuum fully stressed optimization theory is mainly applicable to water tower, chimney, and other tower structures. For high-rise and super high-rise buildings, the constrained effect of floor is small, and the structure mainly appears bending deformation under horizontal load, so the continuity hypothesis is also basically applicable (Barros, et al. 2004; Xu et al., 2017; Yardimoglu, 2010). According to the different structural forms and engineering requirements of high-rise buildings or structures, the equivalent cantilever section forms may be hollow circular, circular, box, and rectangular, etc. The optimal size distribution formulas of circular and rectangular sections can be expressed by the formulas of hollow circular and box section, so the cases of hollow circular and box section are mainly analyzed in this paper. Based on the optimal section size distribution, the optimal shear and bending stiffness distribution are proposed, which can provide reference for practical engineering design and revision of codes.

The optimal section size distribution and the optimal stiffness distribution of engineering structures under different load forms are different. Wind load and earthquake action are common disadvantageous effects for building structures (Cui and Jiang, 2014; Liu et al., 2019; Xu et al., 2018). Equivalent static wind load can be regarded as inverted triangle horizontal load. When only the first mode is considered, the seismic action can be equivalent to the inverted triangle horizontal load; when all modes are considered, the seismic action can be simplified to the uniform load or the inertia force related load (Zhong and Lou, 2016). In order to fully consider the wind load and seismic action, uniform load, inverted triangle load, and inertia force related load are considered in this study.

The fully stressed optimization design requires not only equal stress in the elastic stage, but also equal or similar stress in the elastic-plastic state. At present, in order to simplify the calculation, the constitutive laws of materials are assumed to be linear without nonlinear mechanical properties such as stiffness degradation. However, most of the practical structures are reinforced concrete structures or composite structures. It is difficult to apply the research of linear materials to practice engineering. In order to improve the applicability and feasibility of the research in engineering, homogeneous nonlinear materials are used in the cantilever.

Optimal section size distribution of cantilever with hollow circular variable section

Optimal section size distribution of cantilever under uniform load

A large number of experimental results show that the reinforced concrete in the structure can be equivalent to a composite material with the same properties in tension and compression, and its stress-strain relationship can be characterized by a nonlinear function. In this study, reinforced concrete is equivalent to a composite material. The constitutive relation of the material can be expressed as

σ = f (ε) or ε = g (σ)

(1)

where σ and ε is stress and strain of the material under compression. The complete stress-strain curve equation under compression can be expressed by the concrete polynomial suggested by Hognestad, which is also applicable to reinforced concrete and other composite materials, that is (Hognestad et al., 1955)

σ = a ε^{2} + b ε

(2)

where a and b are undetermined coefficients. The stress-strain curve shows the upward and downward sections, which are the comprehensive response of the actual mechanical properties. In order to simplify the derivation later, it is assumed that the material has the same tensile and compressive properties. The stress-strain curves under tension and compression are the same, that is,

f (- ε) = - f (ε)

(3)

Equation (2) can be expressed as

σ = Sgn (ε) a ε^{2} + b ε

(4)

Firstly, the optimal section size distribution of hollow circular variable section under uniform load is solved. The structure is equivalent to a cantilever with homogeneous nonlinear material. The complete stress-strain curve under compression is expressed by equation (4). The calculation model is shown in Figure 2.

Figure 2.

Mechanical model.

It is assumed that the axis of the cantilever is x axis, the symmetry axis of cross-section is y axis, and the neutral axis is z axis. According to the principle that the bending moment formed by external load is equal to that expressed by the size and bending strain of arbitrary beam section, the equilibrium equation of bending moment is established. According to this principle, the bending strain of the optimized cantilever is equal along the height direction. Because the main strain of the cross-section is bending strain, the Mises strain is approximately equal. In the later FEM, Mises strain of the structure is extracted and the validity of the method is verified. Analytical solution of optimal section size distribution of cantilever is obtained by solving this equation

M_{1} = M_{2} (x)

(5)

where M₁ is the moment formula expressed by the beam section size and strain, and M₂(x) is the moment formula expressed by the external load. M₁ and M₂(x) can be separately solved. When the deformation of the cantilever structure is small, the stain can be expressed by

ε = \frac{y}{ρ} = \frac{2 ε_{1} y}{d} = \frac{2 {ε_{1}}^{'} y}{d'}

(6)

where y is the longitudinal coordinate and ρ is the curvature radius. d and d′ are the external and internal diameter of the hollow circular cross-section. σ₁, σ₂, ε₁, ε₂ and σ′₁, σ′₂, ε′₁, ε′₂ are the stress and strain of the two points farthest from the neutral axis in the external and internal walls, that is, the maximum stress and strain of the external and internal walls in the cross-section. The bending moment of cross-section is expressed as

M_{1} = \int_{A} σ ydA

(7)

where A is area. According to equation (6), equation (7) can be rewrite as

M_{1} = \frac{d^{3}}{4 ε_{1}^{2}} \int_{ε_{2}}^{ε_{1}} \sqrt{1 - {(\frac{ε}{ε_{1}})}^{2}} (a ε^{3} + b ε^{2}) d ε

(8)

Due to the material properties under tension and compression are the same, equation (8) can be expressed by

\begin{matrix} M_{1} = \frac{d^{3}}{2 g^{2} (σ_{1})} \int_{0}^{g (σ_{1})} \sqrt{1 - {(\frac{ε}{g (σ_{1})})}^{2}} (a ε^{3} + b ε^{2}) d ε - \\ \frac{{d'}^{3}}{2 {g'}^{2} (σ_{1})} \int_{0}^{g' (σ_{1})} \sqrt{1 - {(\frac{ε}{g' (σ_{1})})}^{2}} (a ε^{3} + b ε^{2}) d ε \end{matrix}

(9)

Change the variable

ε = g (σ_{1}) \sin t = g' (σ_{1}) \sin u

(10)

Thus, equation (9) becomes

\begin{matrix} M_{1} = \frac{d^{3} g (σ_{1})}{2} \int_{0}^{\frac{π}{2}} \cos^{2} t \sin^{2} t [ag (σ_{1}) \sin t + b] dt - \\ \frac{{d'}^{3} g' (σ_{1})}{2} \int_{0}^{\frac{π}{2}} \cos^{2} u \sin^{2} u [ag' (σ_{1}) \sin u + b] du \end{matrix}

(11)

The ratio of inner diameter to outer diameter is

\frac{d'}{d} = \frac{{ε'}_{1}}{ε_{1}} = \frac{g' (σ_{1})}{g (σ_{1})} = α

(12)

When α is 0, the cross-section is circular. According to equation (12), equation (11) becomes

M_{1} = \frac{d^{3} g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}{480}

(13)

When the cantilever structure is subjected to horizontal loads, the bending moment M₂(x) caused by uniform load q at any height x is

M_{2} (x) = \int_{x}^{H} q (H - x) dx = \frac{q {(H - x)}^{2}}{2}

(14)

where H is total height of the structure. According to equation (5), equations (13) and (14), the optimal section size distribution can be obtained

d (x) = {\frac{240 q {(H - x)}^{2}}{g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{1 / 3}

(15)

Equation (15) is the analytical solution of the optimal section size distribution of a cantilever structure with hollow circular section under uniform load. The maximum stress and strain in each section of the cantilever structure satisfying equation (15) are equal under uniform load.

Optimal section size distribution of cantilever under inverted triangle load

The top load intensity of inverted triangle load is q_max, and the bottom load intensity is 0. When the cantilever is subjected to the inverted triangle load, the moment generated by the load can be obtained

M_{2} (x) = \int_{x}^{H} Q (x) dx = \frac{q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{6 H}

(16)

The bending moment of the cantilever with hollow circular cross-section is equation (13). By establishing the moment equilibrium equation equal to equation (16), the optimal cross-section diameter distribution is

d (x) = {\frac{80 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{Hg (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{1 / 3}

(17)

Equation (17) is the analytical solution of the optimal cross-section size distribution of a cantilever with hollow circular cross-section under inverted triangle load.

Optimal section size distribution of cantilever under inertia force related load

When the cantilever structure with hollow circular cross-section is subjected to the inertia force related load, the load intensity is easily found to be

\bar{q} (x) = λ \bar{m} (x) = \frac{λ ρ π d^{2} (x)}{4}

(18)

where ρ is equivalent surface density and λ is mass correlation coefficient which are used to control the load amplitude. The concentration of the inertia force related load is related to the section area, while the section size is unknown. In order to solve the bending moment caused by load, the following assumptions are made

S ″ (x) = D' (x) = d^{2} (x)

(19)

where D(x) and S(x) are integral and quadratic integral of d(x). The bending moments in the cross-section of a structure at any height x under inertia force related load can be obtained as follows

\begin{matrix} M_{2} (x) = \int_{x}^{H} \frac{λ ρ π [D (H) - D (x)]}{4} dx \\ = \frac{λ ρ π [S (x) - D (H) x + D (H) H - S (H)]}{4} \end{matrix}

(20)

The moment in the hollow circular section can be expressed as equation (13). Optimal section size distribution can be obtained by moment equilibrium equation

d (x) = {\frac{120 λ ρ π [S (x) - D (H) x + D (H) H - S (H)]}{g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{1 / 3}

(21)

Equation (21) is a high-order differential equation of y″ = f(x,y) which belongs to the unsolvable type. Therefore, the method of assuming cross-section size function is considered.

According to uniform displacement criterion, the optimal section size distribution of cantilever was solved by He et al. (2018). The section size distribution function of the continuous cantilever which can realize equal displacement along the height direction under different loads was proposed, and the result was verified by finite element method. The research shows that the distribution of the cross-section size according to the following formula is advantageous to realize the principle of uniform displacement under the action of inertia force related load, and the optimal section distribution function is (He et al., 2018)

h (x) = {(\frac{H + \bar{h} - x}{β})}^{\frac{1}{n}}

(22)

where $\bar{h}$ is the top section radius of cantilever structure. β is the parameter to control the radius of the bottom section and n is the parameter to control the shape of the structure. The computational model is shown in Figure 3.

Figure 3.

Cantilever model under inertia force related load.

If the hollow circular cantilever is subjected to the inertia force related load, the bending moment in the section at any height x is

\begin{matrix} M_{2} (x) = - \frac{λ ρ π β n (1 - α^{2})}{4 (n + 2)} \\ {{(\frac{\bar{h}}{β})}^{\frac{n + 2}{n}} (H - x) + \frac{β n}{2 n + 2} [{(\frac{\bar{h}}{β})}^{\frac{2 n + 2}{n}} - {(\frac{H + \bar{h} - x}{β})}^{\frac{2 n + 2}{n}}]} \end{matrix}

(23)

The moment of the hollow circular section is given by

M_{1} = \frac{{(\frac{H - x}{β})}^{\frac{3}{n}} g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}{480}

(24)

Set equation (24) equals to equation (23), then

\begin{matrix} 32 (1 - α^{5}) a g^{2} (σ_{1}) + 15 (1 - α^{4}) b π g (σ_{1}) + \frac{120 λ ρ π β n (1 - α^{2})}{(n + 2)} {(\frac{H - x}{β})}^{- \frac{3}{n}} \\ {{(\frac{\bar{h}}{β})}^{\frac{n + 2}{n}} (H - x) + \frac{β n}{2 n + 2} [{(\frac{\bar{h}}{β})}^{\frac{2 n + 2}{n}} - {(\frac{H + \bar{h} - x}{β})}^{\frac{2 n + 2}{n}}]} = 0 \end{matrix}

(25)

When $\bar{h}$ = 0 and n = 0.5, equal stress can be realized, and the maximum strain of cross-section may be written as

g (σ_{1}) = \frac{[- 15 (1 - α^{4}) b π + \sqrt{225 {(1 - α^{4})}^{2} b^{2} π^{2} + 512 (1 - α^{5}) (1 - α^{2}) a λ ρ π β^{2}}]}{64 (1 - α^{5}) a}

(26)

Therefore, the optimal cross-section diameter distribution of the cantilever with hollow circular cross-section under inertia force related load is

d (x) = {(\frac{H - x}{β})}^{2}

(27)

Equation (27) is the analytical solution of the optimal cross-section size distribution of a cantilever structure with hollow circular cross-section under inertia force related load.

Optimal section size distribution of cantilever with box variable section

For the research on optimal section size distribution of box section and circular section, the same method is adopted, and the process doesn’t need to be described in detail. The width B of the box external section remains unchanged, and the height h of the external section is optimized. h(x) is a function of section height varying with structural height, and set as

\frac{B'}{B} = μ

(28)

\frac{h'}{h} = \frac{{ε'}_{1}}{ε_{1}} = \frac{g' (σ_{1})}{g (σ_{1})} = γ

(29)

where B′ is width of the internal wall, and h′ is height of the internal wall.

The optimal section size distribution function of box section structure under uniform load is

h (x) = {\frac{12 q {(H - x)}^{2}}{Bg (σ_{1}) [3 (1 - μ γ^{4}) ag (σ_{1}) + 4 (1 - μ γ^{3}) b]}}^{1 / 2}

(30)

The optimal section size distribution function of box section structure under inverted triangle load is

h (x) = {\frac{4 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{HBg (σ_{1}) [3 (1 - μ γ^{4}) ag (σ_{1}) + 4 (1 - μ γ^{3}) b]}}^{1 / 2}

(31)

The optimal section size distribution function of box section structure under inertia force related load is

h (x) = {(\frac{H - x}{β})}^{2}

(32)

Formula simplification under special conditions

In order to satisfy the demand of linear structure optimization design, the result of nonlinear structure is simplified and the design formula of linear structure is put forward. In a later section, the effect of applying the linear structural design formula to the actual nonlinear structural design is verified and compared with the nonlinear structural design formula. The linear constitutive relation of materials can be expressed by

σ = b ε

(33)

The optimal section size distribution functions of hollow circular section linear structure under uniform load, inverted triangle load and inertia force related load are respectively as

d (x) = {[\frac{16 q {(H - x)}^{2}}{g (σ_{1}) (1 - α^{4}) b π}]}^{1 / 3}

(34)

d (x) = {[\frac{16 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{3 g (σ_{1}) (1 - α^{4}) b π H}]}^{1 / 3}

(35)

d (x) = {(\frac{H - x}{β})}^{2}

(36)

The optimal section size distribution functions of box section linear structure under uniform load, inverted triangle load and inertia force related load are

h (x) = {[\frac{3 q {(H - x)}^{2}}{Bg (σ_{1}) (1 - μ γ^{3}) b}]}^{1 / 2}

(37)

h (x) = {[\frac{q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{HBg (σ_{1}) (1 - μ γ^{3}) b}]}^{1 / 2}

(38)

h (x) = {(\frac{H - x}{β})}^{2}

(39)

It will be seen from this that the design formula of linear structure is the same as that of nonlinear structure under inertia force related load.

In practical engineering, there are also design requirements for circular section and rectangular section nonlinear structures. According to the nonlinear structure of hollow circular and box section, the design formula can be obtained. The optimal section size distribution functions of circular section structure under uniform load, inverted triangle load and inertia force related load are as follows

d (x) = {\frac{240 q {(H - x)}^{2}}{g (σ_{1}) [32 ag (σ_{1}) + 15 b π]}}^{1 / 3}

(40)

d (x) = {\frac{80 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{Hg (σ_{1}) [32 ag (σ_{1}) + 15 b π]}}^{1 / 3}

(41)

d (x) = {(\frac{H - x}{β})}^{2}

(42)

The optimal section size distribution functions of rectangular section structure under uniform load, inverted triangle load and inertia force related load are as follows

h (x) = {\frac{12 q {(H - x)}^{2}}{Bg (σ_{1}) [3 ag (σ_{1}) + 4 b]}}^{1 / 2}

(43)

h (x) = {\frac{4 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{HBg (σ_{1}) [3 ag (σ_{1}) + 4 b]}}^{1 / 2}

(44)

h (x) = {(\frac{H - x}{β})}^{2}

(45)

Framework and flowchart of this method

In practical design, first of all, the structural parameters such as section form and external excitation should be determined. Secondly, the material property can be determined and bending moment equilibrium equation should be established. Finally, for continuum structure, structural section size distribution can be obtained according to equations (1) to (45). For building structure, optimal stiffness distribution can provide reference for design. The flowchart of this method is shown in Figure 4.

Figure 4.

Flowchart of this method.

Verification analysis based on finite element method

Nonlinear material constitute

To verify the correctness of the theoretical method and analytical results established in this study, it is necessary to determine the values of specific material nonlinear constitutive model and perform elastic-plastic finite element analysis. The nonlinear constitutive parameters in equation (4) can be determined by fitting the experimental data. There are many axial compression experiments of reinforced concrete columns. In this paper, the parameters are determined according to the data of large-scale reinforced concrete axial compression experiment in Du et al. (2010). In the experiment, eight specimens with different section sizes were studied, each specimen can be used as a fitting without affecting the correctness of the results. The specimen S30–400 is 1800 mm long and the square section size is 400 mm× 400 mm. The slenderness ratio is 4.5. The compressive strength of concrete prism is 22.3 MPa. The yield strength of longitudinal reinforcement is 408.0 MPa, and the equivalent reinforcement ratio of longitudinal reinforcement is 2.51%. The loading system is static loading. The stress-strain curves of S30–400 specimens were fitted by quadratic polynomial to determine the undetermined coefficients a and b. The fitting result of equation (4) is

σ = - 4.6466 \times 10^{6} Sgn (ε) ε^{2} + 2.78 \times 10^{4} ε

(46)

Thus, equation (46) is the expression of stress-strain curves of equivalent composite materials for ordinary reinforced concrete structures under compression. The comparison of experimental data and equation (46) is shown in Figure 5.

Figure 5.

Fitting effect.

Verification for static analysis

In previous studies, the optimal cross-section size formulas of hollow circular and box section nonlinear cantilevers under uniform load, inverted triangle load and inertia force related load are solved. However, the correctness and accuracy of the formulas need to be verified when the structure is in elastic and plastic stages. General finite element software ANSYS is used to analyze the nonlinear cantilever structure with variable cross-section according to the above formulas.

A total of seven cantilever models are established. The total height of the structure is 20 m. The initial modulus of nonlinear materials is 25.94 GPa, and Poisson’s ratio is 0.3. The finite element model has 21 nodes evenly arranged along the height direction of the structure. The bottom of the cantilever is fixed to the ground. One section is set at each node, and the section size is set according to the above formulas. Beam 188 element can be used to simulate Timoshenko beam, considering the effect of shear deformation. So it is adopted in this paper. Taper variable section beam element is set between each adjacent node. According to the analytical solution, the top section area of the cantilever should be 0. However, the ANSYS beam element cannot have a section with an area of 0, so the top section area is slightly larger than 0. The KINH constitutive model is adopt, considering the Bauschinger effect. About 11 data points are set up in the constitutive model, including the upward, peak, and downward sections.

The structural responses of cantilevers with hollow circular and box sections under uniform load, inverted triangle load, and inertia force related load are analyzed. It can be divided into cases of maximum normal strain of cross-section 0.001, 0.002, and 0.003. The design parameters and case conditions are shown in Table 1.

Table 1.

Design parameters and working conditions of cantilevers.

Specimens	Section form	Load form	g(σ₁)
HU1	Hollow circle	Uniform load	0.001
HU2	Hollow circle	Uniform load	0.002
HU3	Hollow circle	Uniform load	0.003
HI1	Hollow circle	Inverted triangle load	0.001
HI2	Hollow circle	Inverted triangle load	0.002
HI3	Hollow circle	Inverted triangle load	0.003
HL1	Hollow circle	Inertia force related load	0.001
HL2	Hollow circle	Inertia force related load	0.002
HL3	Hollow circle	Inertia force related load	0.003
BU1	Box	Uniform load	0.001
BU2	Box	Uniform load	0.002
BU3	Box	Uniform load	0.003
BI1	Box	Inverted triangle load	0.001
BI2	Box	Inverted triangle load	0.002
BI3	Box	Inverted triangle load	0.003
BL1	Box	Inertia force related load	0.001
BL2	Box	Inertia force related load	0.002
BL3	Box	Inertia force related load	0.003

Because the stress and strain in material constitutive law correspond to each other, the strain distribution can be used to characterize the stress distribution. Therefore, Mises strain nephograms of the loaded finite element model is obtained, as shown in Figure 6. It can be seen from the results that the fully stressed state can be achieved in all sections of the continuous cantilever, and it is suitable for the elastic and plastic stages of the structure. The correctness of the key parameters in the formulas are verified. Stress concentration occurs at the top of the structure because a slightly larger area is set at the top of the model. The maximum strain of all elements of the loaded structure is obtained, as shown in Figure 7. The trend of strain nephograms of all components is similar. The strain of the parts below 15 m is less than the design normal strain g(σ₁). The design normal strain is the maximum normal strain of the cross-section under the specified load. The reason is that the extracted strain in FEM is Mises strain. Normal strain of cross-section is the main factor affecting Mises strain. Therefore, although the Mises strain is uniformly distributed, it’s different from the normal strain in value. In addition, the upper and lower sections of taper variable section beam element vary linearly, which is different from the analytical solution of optimal section size distribution.

Figure 6.

Strain nephogram of specimens: (a) HU1, (b) HU2, (c) HU3, (d) HI1, (e) HI2, (f) HI3, (g) BU1, (h) BU2, (i) BU3, (j) BI1, (k) BI2, and (l) BI3.

Figure 7.

Strain distribution of nonlinear specimens.

Under the condition of maximum strain of 0.001, the constant cross-section cantilever without optimization and the optimized structure of variable cross-section cantilever HU1 are compared under uniform load. The bottom diameter of uniform section structure is 8 m, and that of HU1 is 9.38 m. α is 0.9. The parameters of the two finite element models are the same except for the size of cross-section. In addition, due to the phenomenon of stress concentration at the top of HU1, a component with an enlarged top section is designed. The top circular section diameter of it and HU1 are 0.08 and 0.05 m, respectively. The comparison result is shown in Figure 8.

Figure 8.

Comparison of unoptimized structure with optimized structure.

As can be seen from Figure 8, the strain distribution of the unoptimized cantilever structure with equal cross-section is obviously uneven. The strain at the bottom is large, and tends to 0 gradually with the increase of the structural height. Although the strain of the optimized variable cross-section cantilever HU1 has increased nonlinearly since the height of 15 m, there is a certain stress concentration at the top, but the strain below 15 m is almost the same. The volume of uniform section structure is 100.48 m³, the standard deviation of strain distribution is 4.5 × 10⁻⁴. For comparison, the volume of top reinforced structure is 85.56 m³, and the standard deviation of strain distribution is 3.07 e × 10⁻⁶. After optimization, the volume of the structure is reduced by 14.8%, and the strain distribution is more uniform, which fully proves the superiority of the optimization method. It can be seen that the optimized variable cross-section cantilever structure can meet the fully stressed criterion, and the optimization effect is obvious. The phenomenon of stress concentration at the top can be solved by increasing the area of the top section.

Then verify the design formulas of linear structure. Three linear cantilevers with variable cross-section are simulated. The material constitution of linear cantilever can be expressed by equation (33). The other parameters are the same as the finite element model of nonlinear structure. The design parameters and load forms of three linear variable cross-section cantilevers are shown in Table 2.

Table 2.

Design parameters and working conditions of linear cantilever structures.

Specimens	Section form	Load form	g(σ₁)
HUL	Hollow circle	Uniform load	0.001
HLL	Hollow circle	Inertia force related load	0.001
BIL	Box	Inverted triangle load	0.001

The strain along the cantilever height direction is extracted, and the result is shown in Figure 9. It is evident that the optimization results of linear structure are similar to those of nonlinear structure. Hence, the correctness of the design formulas is verified.

Figure 9.

Strain distribution of linear specimens.

It should be pointed out that the structural appearance of the fully stressed criterion is convex under the action of uniform load and inverted triangle load, but it is concave under the inertia force related load. Two types of cantilever structure models are shown in Figure 10. In practical projects, uniform load (inverted triangle load) and inertia force related load may act on the structure at the same time, and they can be given different weights according to different engineering requirements. For example, the optimal section size distribution of hollow circular structure is calculated according to the following formula

\begin{matrix} d (x) = ϕ_{W} {\frac{80 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{Hg (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{1 / 3} \\ + ϕ_{E} {(\frac{H - x}{β})}^{2} \end{matrix}

(47)

where ϕ_W and ϕ_E are the weight coefficients of inverted triangle load and inertia force related load respectively, and ϕ_W + ϕ_E = 1. When ϕ_W = 0.8 and ϕ_E = 0.2, the cantilever model is shown in Figure 10(c).

Figure 10.

Optimized shape based on FEM: (a) for uniform load and inverted triangle load, (b) for inertia force related load, and (c) for uniform load and inertia force related load.

Verification for dynamic analysis

The previous research optimizes the section size distribution of the structure under ground motion and wind load based on statics. In order to verify the accuracy of the optimization results under the dynamic action, the dynamic time history analysis of the cantilever structure with top reinforcement in the previous paper is carried out. The optimized hollow circular and box section cantilevers are compared with the equal section cantilevers, and the unidirectional El Centro seismic wave is input to all structural bases. The time span is 0.02 s, and the duration is 15 s. Since the acceleration time history of seismic wave reaches the peak value at 2.14 s and the displacement time history reaches the peak value at 5.50 s, the strain of each node of the structure at the two times and their four adjacent times is extracted, and the results are shown in Figure 11. The results show that the strain distribution of the hollow circular and box section cantilevers reinforced at the top is uniform under earthquake, so the fully stressed criterion is realized. However, the strain distribution of the unoptimized cantilever decreases nonlinearly from the bottom to the top, and the discreteness of the strain distribution is much greater than that of the optimized cantilever. Through further analysis, it can be confirmed that the strain distribution law of the optimized structure is similar at any time, thus, the fully stressed criterion is completely realized. In summary, the structure optimized according to the inertia force related load realizes the fully stressed criterion under the dynamic action, and the optimization results of different sections are effective.

Figure 11.

Strain distribution of the structures at different time: (a) strain distribution at 2.08 s, (b) strain distribution at 2.14 s, (c) strain distribution at 2.20 s, (d) strain distribution at 5.44 s, (e) strain distribution at 5.50 s, and (f) strain distribution at 5.56 s.

Structural optimal stiffness analysis

The optimal cross-section size distribution formulas of cantilever structures with hollow circular and box sections under uniform load, inverted triangle load and inertia force related load are derived. Based on ANSYS, it is verified that the fully stressed criterion can be realized. The formulas for calculating the shear and bending stiffness of the cantilever are

GA = GA (x)

(48)

EI = EI (x)

(49)

The stiffness corresponding to the formulas under various conditions is calculated according to equations (48) and (49). The results can be used for reference in engineering design and revision of codes.

Under uniform load, the shear, and bending stiffness of the cantilever with hollow circular cross-section are

\frac{G π (1 - α^{2})}{4} {\frac{240 q {(H - x)}^{2}}{g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{2 / 3}

(50)

\frac{E π (1 - α^{4})}{64} {\frac{240 q {(H - x)}^{2}}{g (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{4 / 3}

(51)

Under inverted triangle load, the shear, and bending stiffness of the cantilever with hollow circular cross-section are

\frac{G π (1 - α^{2})}{4} {\frac{80 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{Hg (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{2 / 3}

(52)

\frac{E π (1 - α^{4})}{64} {\frac{80 q_{max} (2 H^{3} - 3 H^{2} x + x^{3})}{Hg (σ_{1}) [32 (1 - α^{5}) ag (σ_{1}) + 15 (1 - α^{4}) b π]}}^{4 / 3}

(53)

Under inertia force related load, the shear, and bending stiffness of the cantilever with hollow circular cross-section are given by

\frac{G π (1 - α^{2})}{4} {(\frac{H - x}{β})}^{4}

(54)

\frac{E π (1 - α^{4})}{64} {(\frac{H - x}{β})}^{8}

(55)

The principle of shear stiffness and bending stiffness of box section cantilever structure is the same, and there is no need to be shown.

The theoretical method and optimization results in this paper are applicable to the static and dynamic optimization design of continuous structures, such as tower structures and chimneys, as well as super high-rise buildings. For a practical engineering structure, it can be designed according to the optimal analytical solution of the size or stiffness of the section. The distribution of lateral stiffness can be flexibly adjusted by adjusting the reinforcement of lateral resisting members and the section size of local members. So the performance parameters of the structure are similar to the theoretical optimization results, and the fully stressed criterion can be realized. Therefore, the optimization results of this study have guiding significance for the actual engineering design.

Practical design example

Solar chimney power plant is a new renewable energy power generation system. It consists of collector and chimney. Schematic of the plant is shown in Figure 12. An unoptimized traditional plant is shown in Figure 13(a). The bottom of the plant is a 20 m diameter collector. The diameter of the chimney is 5 m. The total height is 20 m. An comparative plant is optimized according to equation (27). The appearance of the optimized structure is shown in Figure 13(b). The transition between collector and chimney is smooth. The bottom diameter and total height of the optimized structure is the same as that of the unoptimized structure. In order to realize the function of discharging gas, the cross-section size of chimney above 15 m is enlarged. In order to compare and analyze the structural performance and stress distribution under earthquake excitation, El Centro wave is input to the two models. The strains at 2.08, 2.14, 2.20, 5.44, 5.50, and 5.56 s are extracted and compared as shown in Figure 14. The strain of unoptimized plant decreases from the bottom to the top. The stress concentration at the bottom may cause serious damage to chimney under the earthquake. The strain of optimized structure is almost uniform. Because the cross-section size of chimney above 15 m is enlarged, the strain of the top decrease. But this phenomenon will not reduce the structural safety. The displacement and acceleration time history of the top of structure models is shown in Figure 15. After optimization, the structural dynamic responses obviously decrease. Although the volume of optimized plant is slightly larger than that of unoptimized plant. However, the safety is improved significantly, and the volume can be adjusted by changing the wall thickness. Thus, the effectiveness of the method is verified by practical design example.

Figure 12.

Schematic of solar chimney power plant

Figure 13.

Finite element models of solar chimney power plant: (a) unoptimized structure and (b) optimized structure.

Figure 14.

Strain comparison of the structures at different time: (a) about 2.14 s and (b) about5.50 s.

Figure 15.

Displacement and acceleration time history of the top of structures.

Conclusion

The engineering structures are simplified as cantilever structures with hollow circular and box sections, and the optimization design is carried out with uniform stress objective. The moment equilibrium equation is established, and the analytical solution of the optimal section size is solved. The optimal shear stiffness and bending stiffness distribution are obtained. The following conclusions can be inferred:

For nonlinear structures, the fully stressed criterion can be realized under static load and earthquake. The fully stressed optimization method proposed in this paper is effective. This method can be applied to practical engineering.

The optimization theory of nonlinear structure is also applicable to the linear structure optimization.

Based on ANSYS, the optimal section size formulas of cantilever structures under various conditions are verified respectively. Under static load, the optimized cantilever structure can realize fully stressed state, but stress concentration occurs at the top. This problem can be solved by increasing top section area. Under the earthquake action, the stress of the reinforced cantilever structure is uniform along the height direction.

Under uniform load and inverted triangle load, the cantilever structure optimized according to the uniform stress concept presents the shape of “convex,” while the structure under the inertia force related load presents the shape of “concave.”

In the practical engineering design, the lateral stiffness distribution of the structure can refer to the results proposed in this paper by adjusting the reinforcement and section size of the lateral resisting members, so as to realize the fully stressed criterion. The stiffness distribution proposed in this paper has important guiding significance for the practical engineering design.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China under the Project Number 51878017, which is gratefully acknowledged.

ORCID iD

Haoxiang He

References

Ahlawat

Ramaswamy

(2004) Multiobjective optimal fuzzy logic controller driven active and hybrid control systems for seismically excited nonlinear buildings. Journal of Structal Engineering 130: 416–423.

Ahrari

Deb

(2016) An improved fully stressed design evolution strategy for layout optimization of truss structures. Computers & Structures 164: 127–144.

Barros

MHFM

Barros

AFM

Ferreira

(2004) Closed from solution of optimal design of rectangular reinforced concrete sections. Engineering Computations 16: 66–81.

Bruyneel

Beghin

Craveur

, et al. (2012) Stacking sequence optimization for constant stiffness laminates based on a continuous optimization approach. Structural and Multidisciplinary Optimization 46: 783–794.

Chan

Huang

Kwok

KCS

(2010) Integrated wind load analysis and stiffness optimization of tall buildings with 3D modes. Engineering Structures 32: 1252–1261.

Chan

Sherbourne

Grierson

(1994) Stiffness optimization technique for 3D tall steel building frameworks under multiple lateral loadings. Engineering Structures 16:570–576.

Chan

Zou

(2004) Elastic and inelastic drift performance optimization for reinforced concrete building under earthquake loads. Earthquake Engineering & Structural Dynamics 33: 929–950.

Cui

Jiang

(2014) A morphogenesis method for shape optimization for shape optimization of framed structures subject to spatial constraints. Engineering Structures 77: 109–118.

Dilgen

Aage

, et al. (2019) Topology optimization of acoustic mechanical interaction problems: A comparative review. Structural and Multidisciplinary Optimization 60(2): 779–801.

10.

Zhang

(2010) The experimental study on size effect of the large-size reinforced concrete column under axial loading. China Civil Engineering Journal 43: 1–8 (in Chinese).

11.

Gholizadeh

Mohammadi

(2017) Reliability-based seismic optimization of steel frames by metaheuristics and neural networks. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering 3: 04016013.

12.

Gil-Martín

Aschheim

Hernández-Montes

, et al. (2017) Recent developments in optimal reinforcement of RC beam and column sections. Engineering Structures 26: 101–106.

13.

Guan

Steven

Xie

(1999) Evolutionary structural optimisation incorporating tension and compression materials. Advances in Structural Engineering 2(4): 273–288.

14.

Hajirasouliha

Asadi

Pilakoutas

(2012) An efficient performance-based seismic design method for reinforced concrete frames. Earthquake Engineering & Structural Dynamics 41: 663–679.

15.

Wang

Fan

(2018) Analytical study on optimal stiffness distribution of bend-shear structures considering load distribution. Engineering Mechanics 35: 94–101 (in Chinese).

16.

Hernández-Montes

Gil-Martín

Pasadas-Fernández

(2018) Theorem of optimal reinforcement for reinforced concrete cross sections. Structural and Multidisciplinary Optimization 36: 53–65.

17.

Hognestad

Hanson

McHenry

(1955) Concrete stress distribution in ultimate strength design. ACI J Proc 52: 455–479.

18.

Liu

Jiang

, et al (2019) Stress optimization of smooth continuum structures based on the distortion strain energy density. Computers Methods in Applied Mechanics and Engineering 343: 276–296.

19.

Mergos

(2018) Efficient optimum seismic design of reinforced concrete frames with nonlinear structural analysis procedures. Structural and Multidisciplinary Optimization 58: 2565–2581.

20.

Potra

Simiu

(2009) Optimization and multihazard structural design. Journal of Engineering Mechanics 135: 1472–1475.

21.

San

Waisman

Harari

(2020) Analytical and numerical shape optimization of a class of structures under mass constraints and self-weight. Journal of Engineering Mechanics 146: 04019109.

22.

Truman

Cheng

(1990) Optimum assessment of irregular three-dimensional seismic buildings. Journal of Structal Engineering 116: 3324–3337.

23.

Spencer

, et al. (2017) Optimization of structures subject to stochastic dynamic loading. Computer-Aided Civil & Infrastructure Engineering 32: 657–673.

24.

Xiao

, et al. (2018) Seismic failure mode identification and multi-objective optimization for steel frame structure. Advances in Structural Engineering 21(13): 2005–2017.

25.

Wang

(2005) Optimization and dynamic analysis of frame structures with friction energy dissipation device. Advances in Structural Engineering 8(2): 173–181.

26.

Yardimoglu

(2010) A novel finite element model for vibration analysis of rotating tapered Timoshenko beam of equal strength. Finite Elements in Analysis and Design 46: 838–842.

27.

Zhong

Lou

(2016) Equivalent static wind load on extra-high buildings along wind based on wind-gravity coupling effect. Engineering Mechanics 33: 74–81 (in Chinese).

28.

Zou

Chan

(2005) Optimal seismic performance-based design of reinforced concrete buildings using nonlinear pushover analysis. Engineering Structures 27: 1289–1302.